Subgroup ($H$) information
| Description: | $C_2^6:C_{10}$ |
| Order: | \(640\)\(\medspace = 2^{7} \cdot 5 \) |
| Index: | \(2\) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$\langle(1,3)(8,10), (1,3)(2,6)(4,7)(5,9)(8,10)(12,14), (1,4,8,5,2)(3,7,10,9,6) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_2^6:D_{10}$ |
| Order: | \(1280\)\(\medspace = 2^{8} \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_{16}.C_2^4.C_2^3$ |
| $\operatorname{Aut}(H)$ | $F_{16}.C_4\times \PSL(2,7)$ |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_4\times F_{16}:C_4$, of order \(7680\)\(\medspace = 2^{9} \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2^4:D_5$, of order \(160\)\(\medspace = 2^{5} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_2^3$ | ||||
| Normalizer: | $C_2^6:D_{10}$ | ||||
| Complements: | $C_2$ | ||||
| Minimal over-subgroups: | $C_2^6:D_{10}$ | ||||
| Maximal under-subgroups: | $C_2^5:C_{10}$ | $C_2^5:C_{10}$ | $C_2^5:C_{10}$ | $C_2^7$ | $C_2^2\times C_{10}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_2\wr D_5$ |